Channel estimation using the guard interval of a multicarrier signal

ABSTRACT

A method of communication using Orthogonal Frequency Division Multiplexing (“OFDM”) comprises generating bit streams (b n ∈(0,1),n=0,1, . . . ,K−1) and the corresponding sets of frequency domain carrier amplitudes (X O (k) to X N (k)), where k is the OFDM symbol number, modulated as OFDM symbols to be transmitted from a transmitter. Prefixes are inserted as guard intervals in the sample streams and the OFDM symbols are transmitted from the transmitter to a receiver. The receiver uses information from the prefixes to estimate the Channel Impulse Response (H (F)   D ) of the transmission channels and uses the estimated Channel Impulse Response (Ĥ (F)   D ) to demodulate the bit streams in the signals received. The prefixes (α k .c o  to α k .c D−1 ) are deterministic and are known to the receiver as well as to the transmitter. Preferably, the prefixes (α k .c o  to α k .c D−1 ) comprise a vector (P D ) that is common to said symbols multiplied by at least one weighting factor (α k ). The weighting factor (α k ) preferably differs from one symbol to another but the elements of a given vector (P D ) are multiplied by the same weighting factor. Preferably, the weighting factor (α k ) has a complex pseudo-random value.

FIELD OF THE INVENTION

This invention relates to communication using Orthogonal FrequencyDivision Multiplexing (‘OFDM’) and, more particularly, to channelestimation and tracking in OFDM communication.

BACKGROUND OF THE INVENTION

OFDM communication has been chosen for most of the modern high-data ratecommunication systems (Digital Audio Broadcast—DAB, Terrestrial DigitalVideo Broadcast—DVB-T, and Broadband Radio Access Networks—BRAN such asHIPERLAN/2, IEEE802.11a, for example). However, in most cases thereceiver needs an accurate estimate of the channel impulse response.

In many known OFDM systems, each OFDM symbol of size N∈N⁺ is preceded bya guard interval that is longer than the channel impulse response (CIR)and a cyclic prefix of D∈N⁺ samples is inserted as the guard interval atthe transmitter, the prefix consisting of D samples circularlyreplicated from the useful OFDM symbol time domain samples. The cyclicprefix enables very simple equalisation at the receiver, where thecyclic prefix is discarded and each truncated block is processed, forexample using Fourier Transform (usually Fast Fourier Transform), toconvert the frequency-selective channel output into N parallelflat-faded independent sub-channel outputs, each corresponding to arespective sub-carrier. For equalisation purposes, numerous strategiesexist. Following the zero forcing approach, for example, eachsub-channel output is, unless it is zero, divided by the estimatedchannel coefficient of the corresponding sub-carrier.

Like other digital communication systems, OFDM modulation encountersproblems at high Doppler spreads, which occur notably when the user ismoving fast, for example in a car. HIPERLAN/2, for example, was designedto work only up to speeds of 3 m/s (“pedestrian speed”). Accordingly,the channel impulse response needs to be constantly tracked and updated,especially in the presence of high Doppler spreads.

In a known OFDM communication system pilot tones are added which maychange their position from one OFDM symbol to another. The amplitudesand positions of the pilot tones are known to the receiver. The receiveruses the pilot tones to estimate the channel coefficients of thecorresponding carriers. This method is widely used, but it degrades thesystem performance, since a certain number of carriers cannot be usedfor data, since they are reserved for the pilot tones.

It is also known to add learning sequences (See for example EBU ReviewTechnical No. 224, August 1987, “Principles of modulation and channelcoding for digital broadcasting for mobile receiver”, by M. Alard and R.Lassalle.). In HIPERLAN/2, for example, there are at least 2 learningOFDM symbols per frame (i.e. 2 OFDM symbols of 2·4 μs duration in totalper 2 ms). If the channel changes quickly, there must be many moretraining sequences and the consequence is an even bigger degradation inthe system performance.

Many of the known systems are unable to decode all carriers of OFDMsymbols in the presence of channel nulls. Recent innovations proposeways for decoding OFDM symbols even in the presence of channel nulls(see for example the publication entitled “Reduced Complexity Equalizersfor Zero-Padded OFDM transmissions” by B. Muquet, Marc de Courville, G.B. Giannakis, Z. Wang, P. Duhamel in the proceedings of theInternational Conference on Acoustics Speech and Signal Processing(‘ICASSP’) 2000 and the publication entitled “OFDM with trailing zerosversus OFDM with cyclic prefix: links, comparisons and application tothe HiperLAN/2 system” by Muquet, B.; de Courville, M.; Dunamel, P.;Giannakis, G. in the proceedings of the IEEE International Conference onCommunications, 2000, Volume: 2. However, these publications do notoffer responses to the problems referred to above concerning channelestimation and channel tracking.

Ideally, the OFDM modulation system would keep all the advantages ofclassic OFDM and additionally allow very simple and completely blindchannel estimation at the receiver. No additional redundancy would beadded to the system and therefore no bandwidth would be lost. Such asystem would be advantageous in low-mobility scenarios and would makeOFDM systems applicable to high-mobility scenarios as well.

Many of the examples and illustrations presented below are based on theassumption N=4·D, that is to say that the size of the prefix (D samples)is assumed to be one quarter of the size of the useful OFDM symbol (Nsamples). This corresponds to the case of HiperLAN/2 or IEEE802.11. Thisrestriction is introduced for sake of simplicity only. It will beappreciated that the examples and illustrations are applicable moregenerally to the case of N∈N⁺, D∈N⁺, the necessary adaptation beingbasically straightforward.

SUMMARY OF THE INVENTION

The present invention provides a method of, and a transmitter and areceiver for, communication using OFDM as described in the accompanyingclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block schematic diagram of an OFDM communication systemcomprising a transmitter and a receiver in accordance with oneembodiment of the invention, given by way of example,

FIG. 2 is a schematic diagram of an OFDM frame in a signal appearing inoperation of the system of FIG. 1,

FIG. 3 is a matrix equation representing the channel impulse responsefor inter-block interference in operation of the system of FIG. 1,

FIG. 4 is a matrix equation representing the channel impulse responsefor inter-symbol interference in operation of the system of FIG. 1,

FIG. 5 is a matrix equation representing the combined channel impulseresponse in operation of the system of FIG. 1,

FIG. 6 is a representation of a sub-matrix corresponding to the combinedchannel impulse response in operation of the system of FIG. 1 for aprefix part of the signal of FIG. 2,

FIG. 7 represents the upper triangular sub-matrix of the channel matrixpresented by FIG. 6,

FIG. 8 represents the lower triangular sub-matrix of the channel matrixpresented by FIG. 6, and

FIG. 9 is a matrix equation representing signals appearing as a resultof the combined channel impulse response in operation of one embodimentof a system of the kind shown in FIG. 1,

FIG. 10 is a matrix equation representing signals appearing duringchannel estimation in operation of one embodiment of a system as shownin FIG. 1,

FIG. 11 is a matrix equation representing signals appearing as a resultof the combined channel impulse response in operation of anotherembodiment of a system of the kind shown in FIG. 1

FIG. 12 is a graph representing preferred values of prefixes used in thesystem of FIG. 1.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 shows an OFDM communication system in accordance with oneembodiment of the invention comprising a transmitter comprising an OFDMmodulator 1 and a receiver comprising an OFDM demodulator 2, thetransmitter and the receiver communicating over a communication channel3.

An input bit-stream b_(n)∈(0,1),n=0,1, . . . ,K−1 is modulated onto aset of N carriers whose carrier amplitudes are given by the vectorX(k)=(X₀(k),X₁(k), . . . ,X_(N−1)(k))^(T), corresponding to OFDM symbolnumber k. Afterwards, the time domain OFDM signal is generated by means4 which performs an Inverse Fourier Transform operation, or preferablyan Inverse Fast Fourier Transform (‘IFFT’) operation[F_(N)]⁻¹=[F_(N)]^(H) with [F_(N)]^(H)=([F_(N)]^(T))* where (·)^(T) isthe transposition operator and (·)* is the complex conjugate operator:$\begin{matrix}{{{x(k)} = {{\left\lbrack F_{N} \right\rbrack^{- 1}{X(k)}} = {\left( {{x_{0}(k)},{x_{1}(k)},\ldots\quad,{x_{N - 1}(k)}} \right)^{T}.}}},{{{where}\left\lbrack F_{N} \right\rbrack} = {\frac{1}{\sqrt{N}} \cdot \left( W_{N}^{lk} \right)_{{0 \leq l \leq {N - 1}},{0 \leq k \leq {N - 1}}}}},{{{and}\quad W_{N}} = {{\mathbb{e}}^{{- j}\frac{2\quad\pi}{N}}.}}} & {{Equation}\quad 1}\end{matrix}$

The resulting parallel signal x (k) vector is converted to a seriessignal by a parallel-to-series converter 5, a prefix, represented by theD×1 vector P_(D)=(c₀, . . . ,c_(D−1))^(T), being inserted into thesignal as guard interval between each OFDM symbol to produce a seriesdigital signal x_(n). The series digital signal x_(n) is then convertedto an analogue signal x(t) by a digital-to-analogue converter 6 andtransmitted over the channel 3.

The channel 3 has a Channel Impulse Response H(k)=C(k) and alsointroduces noise v.

At the receiver 2, an analogue signal r(t) is received and converted toa digital signal r_(n) by an analogue-to-digital converter 7. Thedigital signal r_(n) is then converted to a parallel signal by aseries-to-parallel converter r(k) and equalised and demodulated byequalisation and demodulation means 9 to produce demodulated signalss^(est)(k). In the following analysis, consideration of noise is omittedfor the sake for simplicity. However, including the consideration ofnoise does not significantly modify the results.

In some known OFDM communication systems, the guard interval is used toadd some redundancy (D samples of redundancy are added) by introducing acyclic prefix, for example in the following manner:x^((CP))(k)=(x_(N−D)(k), . . . ,x_(N−1)(k),x₀(k),x₁(k), . . .,x_(N−1)(k))^(T).In other words, data from the end of the frame is repeated by thetransmitter in the guard interval to produce a prefix.

In accordance with this embodiment of the present invention, however,the prefix samples inserted as guard interval of OFDM symbol number k,α_(k).c₀ to α_(k).c_(D−1), are deterministic and are known to saidreceiver as well as to said transmitter. The prefixes comprise a vectorP_(D)=(c₀, . . . ,c_(D−1))^(T) of size D×1 that is common to the symbolsmultiplied by at least one weighting factor α_(k), so that the prefixeshave the overall form α_(k).c₀ to α_(k).c_(D−1). The weighting factorα_(k) may be constant from one symbol to another. However, in apreferred embodiment of the invention, the weighting factor ak differsfrom one symbol to another, the elements of a given vector P_(D) beingmultiplied by the same weighting factor. With an OFDM modulator in thetransmitter functioning in this way, blind channel estimation in thereceiver can be done simply and at low arithmetical complexity. Inparticular, the receiver can constantly estimate and track the channelimpulse response without any loss of data bandwidth. Moreover, thedemodulator at the receiver can have advantageous characteristics,ranging from very low arithmetical cost (at medium performance) to higharithmetical cost (very good system performance).

More particularly, in the preferred embodiment of the invention, theprefix of D samples that is added in the guard interval comprises apre-calculated suitable vector P_(D)=(c₀, . . . ,c_(D−1))^(T) of Dsamples that is independent of the data and that is weighted by apseudo-random factor α_(k) that only depends on the number k of thelatest OFDM symbol:x^((const))(k)=(α_(k)c₀, . . . ,α_(k)c_(D−1),x₀(k),x₁(k), . . .,x_(N−1)(k))^(T).  Equation 2

For the purposes of the analysis below, a second prefix/OFDM symbolvector is defined as follows:x^((const,post))(k)=(x₀(k),x₁(k), . . . ,x_(N−1)(k),α_(k+1)c₀, . . .,α_(k+1)c_(D−1))^(T).  Equation 3

Several choices for α_(k) are possible. It is possible to chooseα_(k)∈C, that is to say that α_(k) can be of any complex value. However,any α_(k) with |α_(k)|≠1 leads to performance degradation compared topreferred embodiments of the invention.

It is possible to limit the choice of α_(k), somewhat less generally toα_(k)∈C with |α_(k)|=1. This choice usually leads to good systemperformance, but the decoding process risks to be unnecessarily complex.

Accordingly, in the preferred embodiment of the present invention, thephase of α_(k) is chosen so that${\alpha_{k} = {\mathbb{e}}^{j{\frac{2\quad\pi}{N + D} \cdot m}}},$where m is an integer, N is the useful OFDM symbol size and D is thesize of the pseudo-random prefix. This choice is particularlyadvantageous when using the specific decoding methods described below.

For the sake of simplicity, the following analysis assumes that theweighting factor has been chosen as${\alpha_{k} = {\mathbb{e}}^{j{\frac{2\quad\pi}{N + D} \cdot m}}},$integer. However, it will be appreciated that the mathematicaladaptation to any of the cases presented above is straightforward.

It proves to be very useful to choose ak such that its phase changesfrom OFDM symbol to OFDM symbol. The constant prefix P_(D) is preferablychosen with respect to-certain criteria, for example the following:

-   -   In the frequency domain, P_(D) is as flat as possible over the        frequency band used for data carriers.    -   In the frequency domain, P_(D) is as near to zero as possible        for all unused parts of the band.    -   In the time domain, P_(D) has a low peak-to-average-power-ratio        (PAPR).    -   The length of P_(D) is the size of the OFDM guard interval, that        is to say D samples. Alternatively, a shorter sequence of length        {circumflex over (D)}<D may be chosen where D−{circumflex over        (D)} zeros are appended.

With these criteria, without any complication of the transmitter, thereceiver is able to estimate the channel impulse response blindly, trackthe changes of the channel impulse response blindly and perform anarithmetically simple equalization.

An example of a frame of OFDM symbols in accordance with a preferredembodiment of the invention is illustrated in FIG. 2. The operation ofthe system will first be described for the specific case where α_(k) isconstant and equal to 1.

Now, the modulation unit of the transmitter is clearly defined. In thefollowing, the operations to be performed in the receiver areconsidered. Each received OFDM symbol selected at the input of thedemodulator 9 can then be expressed as follows (neglecting additivenoise):r(k)=[H _(IBI) ]·x ^((const,post))(k−1)+[H _(ISI) ]·x^((const,post))(k).  Equation 4where the channel impulse response of the demodulator 9 is assumed to beh=(h₀, . . . ,h_(D−1)), [H_(IBI)] is the contribution of the demodulator9 channel matrix corresponding to inter-block-interference and [H_(ISI)]is its contribution to inter-symbol-interference.

The components of the received signal corresponding tointer-block-interference [H_(IBI)]r(k−1) are illustrated in FIG. 3,where blank elements correspond to zero values, for an example whereN=4·D (for example, in the case of HiperLAN/2 or IEEE802.11, N=64 andD=16). It will be seen that [H_(IBI)] is a matrix of size (N+D)×(N+D)with a triangular sub-matrix [H₁] of size (D−1)×(D−1) at its upperright-hand comer, illustrated by FIG. 7, the other elements of thematrix being zero.

The components of the received signal corresponding tointer-symbol-interference [H_(ISI)]r(k−1) are illustrated in FIG. 4, forthe same case and in the same manner as FIG. 3. It will be seen that[H_(ISI)] is a matrix of size (N+D)×(N+D) with triangular sub-matrices[H₁] on its major diagonal as illustrated by FIG. 7 and triangularsub-matrices [H₀] of size D×D on the diagonal immediately below the maindiagonal as illustrated by FIG. 8, the other elements of the matrixbeing zero.

The channel impulse response seen by demodulator 9 is represented by thesum of the inter-block-interference [H_(IBI)] and theinter-symbol-interference [H_(ISI)], as shown in FIG. 5. The resultingsignal for this example is shown in FIG. 9, where r₀(k) to r₄(k) aresuccessive parts of the OFDM symbol #k containing as well contributionsof the preceding and following prefix convolved by the channel, x₀(k) tox₃(k) are corresponding parts of size D of the useful signal transmittedand x₄(k) is a corresponding part of size D of the following prefix inthis example. Of course, the example may be generalised to any N∈N⁺,D∈N⁺.

The expectation values of the parts of the received signals are asfollows: $\begin{matrix}\begin{matrix}{E_{0} = {E\left( r_{0} \right)}} \\{= {\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{0} \right\rbrack \cdot {x_{0}(k)}} \right)} + {E\left( {\left\lbrack H_{1} \right\rbrack \cdot {x_{4}(k)}} \right)}}} \\{= {\left\lbrack H_{1} \right\rbrack \cdot P_{D}}}\end{matrix} & {{Equation}\quad 5} \\\begin{matrix}{E_{1} = {E\left( r_{1} \right)}} \\{= {{\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{1} \right\rbrack \cdot {x_{0}(k)}} \right)} + \underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{0} \right\rbrack \cdot {x_{1}(k)}} \right)}} = 0}}\end{matrix} & {{Equation}\quad 6} \\\begin{matrix}{E_{2} = {E\left( r_{2} \right)}} \\{= {{\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{1} \right\rbrack \cdot {x_{1}(k)}} \right)} + \underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{0} \right\rbrack \cdot {x_{2}(k)}} \right)}} = 0}}\end{matrix} & {{Equation}\quad 7} \\\begin{matrix}{E_{3} = {E\left( r_{3} \right)}} \\{= {{\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{1} \right\rbrack \cdot {x_{2}(k)}} \right)} + \underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{0} \right\rbrack \cdot {x_{3}(k)}} \right)}} = 0}}\end{matrix} & {{Equation}\quad 8} \\\begin{matrix}{E_{4} = {E\left( r_{4} \right)}} \\{= {\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{1} \right\rbrack \cdot {x_{3}(k)}} \right)} + {E\left( {\left\lbrack H_{0} \right\rbrack \cdot {x_{4}(k)}} \right)}}} \\{= {\left\lbrack H_{0} \right\rbrack \cdot P_{D}}}\end{matrix} & {{Equation}\quad 9}\end{matrix}$

It will be appreciated that the expectation values of the useful partsx₀(k) to x₃(k) of the OFDM symbol tend to zero over a large number ofsymbols since they are quasi-random with zero mean. However, the prefixP_(D) is known to the receiver (and in this embodiment is constant oversuccessive symbols) and enables └H ┘=[H₀]+[H₁] to be estimated, byapproximating the expectation values E₀ and E₄ over a large number R ofsymbols: $\begin{matrix}{{E(r)} \approx {\frac{1}{R} \cdot {\sum\limits_{l = 0}^{R - 1}{{r(l)}.}}}} & {{Equation}\quad 10}\end{matrix}$

The sum of the expectation values E₀ and E₄ is then given by:E ₀ +E ₄=([H ₀ ]+[H ₁])·P _(D) =[H]·P _(D)  Equation 11

A first embodiment of a method of channel impulse response estimation onD symbols in accordance with the present invention utilises theexpression of the above equation as follows: $\begin{matrix}\begin{matrix}{{{\cdot P_{D}} + {\cdot P_{D}}} = {H \cdot P_{D}}} \\{= {\left\lbrack F_{D} \right\rbrack^{- 1} \cdot {{diag}\left( {H_{0},H_{1},\ldots\quad,H_{D - 1}} \right)} \cdot}} \\{\left\lbrack F_{D} \right\rbrack \cdot P_{D}}\end{matrix} & {{Equation}\quad 12}\end{matrix}$where the matrices [F_(D)] and [F_(D)]⁻¹=[F_(D)]^(H)=(F_(D) ^(T))* arethe (Fast) Fourier Transform and Inverse (Fast) Fourier Transformmatrices respectively and the prefix P_(D) is of size D. The matrices[H₀],[H₁] and [H] are illustrated by FIG. 7, FIG. 8 and FIG. 6respectively.

Accordingly, in this first method, the channel impulse response isestimated using the following steps:

-   Perform a FFT_(D×D) on V_(HP)=([H₀]+[H₁])·P_(D)=E₀+E₄-   Perform a FFT_(D×D) on V_(P)=P_(D)-   Perform a component-by-component division of the first result by the    second Ĥ_(D) ^((F))=V_(HP)    V_(P):    Ĥ _(D) ^((F)) =FFT _(D×D)(([H ₀ ]+[H ₁])·P _(D))    FFT _(D×D)(P _(D)).-   Perform an IFFT on Ĥ_(D) ^((F)):    ĥ_(D)=FFT_(D×D) ⁻¹(Ĥ_(D) ^((F))).

The resulting channel estimation is ĥ_(D) of size D×1. This method workswell in many circumstances and has a low arithmetic cost, since itscalculations are based on matrices of size D×D. However, an OFDM symbolwhich usually is of size N>D samples will be equalized based on thisestimation. Thus, this method works very well if the prefix-spectrum isnon-zero everywhere in the FFT_(D×D) domain (and, of course, everywherewell above channel noise). This can be a troublesome limitation in othercircumstances.

A second embodiment of a method of channel impulse response estimationon D carriers in accordance with the present invention avoids thislimitation, at the expense of increased arithmetic cost. This secondmethod does not estimate ĥ_(D) based on a de-convolution in theFFT_(D×D) domain as presented above, but estimatesFFT_((N+D)×(N+D))((ĥ_(D) ^(T)0_(N) ^(T))^(T)) directly based on thereceived vector ([H₀]+[H₁])·P_(D). This is possible by exploiting theobservation:[H _((N+D)×(N+D)]·() P _(D) ^(T)0_(N) ^(T))^(T)=(E ₄ ^(T) E ₀^(T)0_(N−D) ^(T))^(T)  Equation 13

This equation is represented in more detail in FIG. 10. In this secondmethod, the channel impulse response is estimated using the followingsteps:

-   Perform a FFT_((N+D)×(N+D)) on V_(HP)=[H_((N+D)×(N+D)]·(P) _(D)    ^(T)0_(N) ^(T))^(T)-   Perform a FFT_((N+D)×(N+D)) on V_(P)=(P_(D) ^(T)0_(N) ^(T))^(T)-   Perform a component-by-component division Ĥ_(N+D) ^((F))=V_(HP)    V_(P)-   If desired, perform an IFFT on Ĥ_((N+D)) ^((F)):    ĥ_((N+D))=IFFT_((N+D)×(N+D))(Ĥ_(N+D) ^((F)))

The last step of the list presented above is not essential for the basicequalization algorithm but may be useful, for example in algorithms usedto reduce noise levels.

The above methods have been described with reference to the specificcase where α_(k) is constant and equal to 1. In preferred embodiments ofthe invention, however, the weight α_(k) of the prefix to each symbol kis a preferably complex pseudo-random factor that only depends on thenumber k of the latest OFDM symbol. The adaptations to this method ofthe basic equations (shown in FIG. 9) are shown in FIG. 11.

It is found that equations 4 and 8 are to be adapted as follows:$\begin{matrix}\begin{matrix}{E_{\alpha,0} = {\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{0} \right\rbrack \cdot {r_{0}(k)}} \right)} + {E\left( {\left\lbrack H_{1} \right\rbrack \cdot \frac{\left( {\alpha_{k} \cdot P_{D}} \right)}{\alpha_{k}}} \right)}}} \\{= {\left\lbrack H_{1} \right\rbrack \cdot {P_{D}.}}}\end{matrix} & {{Equation}\quad 14} \\\begin{matrix}{E_{\alpha,4} = {\underset{\underset{= 0}{︸}}{E\left( {\left\lbrack H_{1} \right\rbrack \cdot {r_{3}(k)}} \right)} + {E\left( {\left\lbrack H_{0} \right\rbrack \cdot \frac{\left( {\alpha_{k + 1} \cdot P_{D}} \right)}{\alpha_{k + 1}}} \right)}}} \\{= {\left\lbrack H_{0} \right\rbrack \cdot {P_{D}.}}}\end{matrix} & {{Equation}\quad 15}\end{matrix}$

The procedures for blind channel estimation described above remainapplicable by setting E₀=E_(α,0) and E₄=E_(α,4). This amounts toweighting the preceding and following D prefix-samples of each receivedsymbol by the corresponding α_(k) ⁻¹ or α_(k+1) ⁻¹, respectively.

The values of the prefixes α_(k)·P_(D) are chosen as a function ofselected criteria, as mentioned above. Values that have been found togive good results with the criteria:

-   Low Peak-to-Average-Power-Ratio of the time domain signal-   Low Out-of-Band Radiation, that is to say maximise the energy of the    prefix over the useful band and not waste prefix energy over null    carriers-   Spectral Flatness, e.g. SNR of each channel estimates shall be    approx. constant-   Low-Complexity Channel Estimation, i.e. by prefix spectrum whose    spectral contributions are mainly just phases (i.e. of constant    modulus),    are shown in FIG. 12 by way of example, for the following OFDM    parameters:-   Size of the Prefix in Time Domain: D=16 Samples-   Size of the OFDM symbols in the frame: N=64 Samples-   Carriers where channel coefficients are to be estimated (over N+D=80    carriers): Carriers 1 to 52-   Out-of-Band region: Carriers 76 to 80-   Maximum PAPR has not been limited-   Out-of-Band Radiation as low as possible-   Spectral Flatness as good as possible.

The channel estimation is done by calculating the expectation value overa number of samples of the received vector as explained above. If thetracking of the channel is done based on a first estimationĥ(k−1)=[F_(D)]^(H)·Ĥ(k−1) of the channel impulse response and a number Rof OFDM symbols, the first estimate is then updated as follows:${\hat{H}(k)} = {{s_{0} \cdot {\hat{H}\left( {k - 1} \right)}} + {\left( {\left\lbrack F_{D} \right\rbrack \cdot {\sum\limits_{n = 0}^{R - 1}{s_{n + 1} \cdot \left( {{E_{0}\left( {k - n} \right)} + {E_{4}\left( {k - n} \right)}} \right)}}} \right)\left( {\left\lbrack F_{D} \right\rbrack \cdot P_{D}} \right)}}$ĥ(k) = [F_(D)]^(H)Ĥ(k).based on the ideas of the first method for channel estimation that hasbeen presented above. Alternatively, the second method can be applied by${\hat{H}(k)} = {{s_{0} \cdot {\hat{H}\left( {k - 1} \right)}} + {\left( {\left\lbrack F_{N + D} \right\rbrack \cdot {\sum\limits_{n = 0}^{R - 1}{s_{n + 1} \cdot \left( {{E_{4}^{T}\left( {k - n} \right)},{E_{0}^{T}\left( {k - n} \right)},0_{N - D}^{T}} \right)^{T}}}} \right)\left( {\left\lbrack F_{N + D} \right\rbrack \cdot \left( {P_{D}^{T},0_{N}^{T}} \right)^{T}} \right)}}$ĥ(k) = [F_(N + D)]^(H)Ĥ(k).where the factors s_(n),n=0,1, . . . ,R−1 are positive real numbers thatare used for normalization and weighting of the different contributions.Thus, for example it is possible to take older OFDM symbols less intoaccount for the channel estimation than later ones. The Fourier matrix[F] can be chosen in the N+D carriers or D carriers domain

Several equalization methods are advantageous using the pseudo-randomprefix OFDM. In general, the different methods offer differentperformance-complexity trade-offs.

A first embodiment of a method of equalization uses zero forcing in theN+D Domain and offers low complexity equalization.

With ${\beta_{k} = \frac{\alpha_{k}}{\alpha_{k + 1}}},$the Channel Impulse Response matrix can be represented as follows:$\begin{matrix}\begin{matrix}{\lbrack H\rbrack = {{\left\lbrack H_{ISI} \right\rbrack + {\frac{\alpha_{k}}{\alpha_{k + 1}} \cdot \left\lbrack H_{IBI} \right\rbrack}} = {\left\lbrack H_{ISI} \right\rbrack + {\beta_{k} \cdot \left\lbrack H_{IBI} \right\rbrack}}}} \\{= \left\lbrack \left. \begin{matrix}h_{0} & {\beta_{k} \cdot h_{N + D - 1}} & {\beta_{k} \cdot h_{N + D - 2}} & \ldots & \ldots & {\beta_{k} \cdot h_{1}} \\h_{1} & h_{0} & {\beta_{k} \cdot h_{N + D - 1}} & \ldots & \ldots & {\beta_{k} \cdot h_{2}} \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\h_{N + D - 1} & \ldots & \ldots & \ldots & \ldots & h_{0}\end{matrix} \right| \right\rbrack}\end{matrix} & {{Equation}\quad 16}\end{matrix}$

Still assuming that the length of the channel impulse response is D, thecoefficients h_(D+1), . . . , N+D−1 are set to zero. This is a so-calledpseudo-circulant matrix, corresponding to the case where β^(k) is notequal to 1, and can be diagonalized as follows: $\begin{matrix}{{\lbrack H\rbrack = {{\frac{1}{\sqrt{N + D}} \cdot \lbrack V\rbrack^{- 1} \cdot {{diag}\left( {{H\left( \beta_{k}^{\frac{1}{N + D}} \right)},{H\left( {{\mathbb{e}}^{j\quad \cdot \frac{2 \cdot \pi}{N + D}} \cdot \beta_{k}^{\frac{1}{N + D}}} \right)},\ldots\quad,{H\left( {{\mathbb{e}}^{{j\quad \cdot \frac{2 \cdot \pi}{N + D}}{({N + D - 1})}} \cdot \beta_{k}^{\frac{1}{N + D}}} \right)}} \right)} \cdot \lbrack V\rbrack \cdot {\sqrt{N + D}.{{where}\lbrack V\rbrack}}} = {{\left( {\sum\limits_{n = 0}^{N + D - 1}{\beta_{k}}^{\frac{2n}{N + D}}} \right)^{\frac{1}{2}} \cdot \left\lbrack F_{N + D} \right\rbrack \cdot {diag}}\left\{ {1,\beta_{k}^{\frac{1}{N + D}},\ldots\quad,\beta_{k}^{\frac{N + D - 1}{N + D}}} \right\}}}}{{{and}\quad{H(z)}} = {\sum\limits_{n = 0}^{N + D - 1}{z^{- n} \cdot h_{n}}}}} & {{Equation}\quad 17}\end{matrix}$

With the assumption that r(k)=[H]·(x(k)^(T)P_(D) ^(T))^(T), and that theweighting factor has been chosen as${\beta_{k} = {\frac{\alpha_{k}}{\alpha_{k + 1}} = {\mathbb{e}}^{j{\frac{2\quad\pi}{N + D} \cdot m}}}},$where m is an integer, and if the received vector R(k) is:R(k)=[H _(α) _(k) _(,(N+D)×(N+D))]·(x(k)^(T) P _(D) ^(T))^(T),the procedure of this method of zero forcing equalisation is:

-   Perform Multiplication R⁽¹⁾(k)=√{square root over (N+D)}·[V]·R(k),    where    ${\lbrack V\rbrack = {{\left( {\sum\limits_{n = 0}^{N + D - 1}{\beta_{k}}^{\frac{2n}{N + D}}} \right)^{\frac{1}{2}} \cdot \left\lbrack F_{N + D} \right\rbrack \cdot {diag}}\left\{ {1,\beta_{k}^{\frac{1}{N + D}},\ldots\quad,\beta_{k}^{\frac{N + D - 1}{N + D}}} \right\}}},$-   Calculate the frequency shifted, estimated CIR coefficients    ${\hat{H}}_{N + D}^{{Shifted},F} = {\left( {{\hat{H}\left( \beta_{k}^{\frac{1}{N + D}} \right)},\ldots\quad,{\hat{H}\left( {\beta_{k}^{\frac{1}{N + D}} \cdot {\mathbb{e}}^{j\quad 2\quad\pi\frac{N + D - 1}{N + D}}} \right)}} \right).}$-   Perform a component-by-component division R⁽²⁾(k)=R⁽¹⁾(k)    Ĥ_(N+D) ^(Shifted,F)-   Perform Multiplication    ${R^{(3)}(k)} = {\frac{1}{\sqrt{N + D}} \cdot \lbrack V\rbrack^{- 1} \cdot {{R^{(2)}(k)}.}}$-   Extract the N equalized samples of the kth OFDM-data symbol to    S^(EQ)(k).-   Transform the kth OFDM data symbol S^(EQ)(k) into the frequency    domain by a Fourier transform S_(F) ^(EQ)(k)=[F_(N)]·S^(EQ)(k).-   Proceed with metric calculation, etc. on the received equalised    carriers.

Another embodiment of a method of equalization uses a method known fromstudies on zero padding. The received vector R(k) in the OFDM PseudoRandom Prefix Scheme can be expressed as follows, where [P] contains a(N+D)×N pre-coding matrix and I_(N) is the N×N identity matrix:$\begin{matrix}\begin{matrix}{{R(k)} = {{\lbrack H\rbrack \cdot \left( {{\lbrack P\rbrack \cdot {x(k)}} + \begin{bmatrix}0_{N \times N} \\{\alpha_{k + 1} \cdot P_{D}}\end{bmatrix}} \right)} + v}} \\{= {{\lbrack H\rbrack \cdot \left( {{\begin{bmatrix}I_{N} \\0_{D \times N}\end{bmatrix} \cdot {x(k)}} + \begin{bmatrix}0_{N \times N} \\{\alpha_{k + 1} \cdot P_{D}}\end{bmatrix}} \right)} + v}}\end{matrix} & {{Equation}\quad 18}\end{matrix}$

The Channel Impulse Response estimation obtained as described above isthen used together with the known values of P_(D) to perform thefollowing operation $\begin{matrix}\begin{matrix}{{R^{(1)}(k)} = {{R(k)} - {\left\lbrack \hat{H} \right\rbrack \cdot \begin{bmatrix}0_{N \times N} \\{\alpha_{k + 1} \cdot P_{D}}\end{bmatrix}}}} \\{= {{\lbrack H\rbrack \cdot \begin{bmatrix}I_{N} \\0_{D \times N}\end{bmatrix} \cdot {x(k)}} + v}}\end{matrix} & {{Equation}\quad 19}\end{matrix}$in which the known prefix values are multiplied by the Channel ImpulseResponse estimation and the result subtracted from the received signal.In the general case, [H] is a pseudo circulant channel matrix. So, thediagonalisation of such matrices can then be performed in order tocalculate [H]·P_(D) efficiently. Then, several equalization approachesare possible, for example the Zero Forcing (ZF) approach or the MinimumMean Square Error (MMSE) equalization approach. Examples of MMSEequalization methods are described in the articles “OFDM with trailingzeros versus OFDM with cyclic prefix: links, comparisons and applicationto the HiperLAN/2 system” by Muquet, B.; de Courville, M.; Dunamel, P.;Giannakis, G. ICC 2000 - IEEE International Conference onCommunications, Volume 2, 2000 and “Reduced Complexity Equalizers forZero-Padded OFDM transmissions” by Muquet, B.; de Courville, M.;Giannakis, G. B.; Wang, Z.; Duhamel, P. International Conference onAcoustics Speech and Signal Processing (ICASSP) 2000.

In one example, the equalisation is performed based on a zero-forcingapproach by multiplying y⁽¹⁾ by the Moore-Penrose pseudo-inverse [G] ofthe matrix ${{\lbrack H\rbrack \cdot \begin{bmatrix}I_{N} \\0_{D \times N}\end{bmatrix}}{\text{:}\lbrack G\rbrack}} = {\left\lbrack {\lbrack H\rbrack \cdot \begin{bmatrix}I_{N} \\0_{D \times N}\end{bmatrix}} \right\rbrack^{+}.}$Thus, the equalized resulting vector is $\begin{matrix}\begin{matrix}{{R^{({{eq},{ZF}})}(k)} = {\lbrack G\rbrack \cdot {R^{(1)}(k)}}} \\{= {\lbrack G\rbrack \cdot {\left\lbrack {{\lbrack H\rbrack \cdot \begin{bmatrix}I_{N} \\0_{D \times N}\end{bmatrix} \cdot {x(k)}} + v} \right\rbrack.}}}\end{matrix} & {{Equation}\quad 20}\end{matrix}$

The definition of the Moore-Penrose pseudo-inverse is, among others,discussed by Haykin in the book: “Adaptive Filter Theol” by SimonHaykin, 3^(rd) edition, Prentice Hall Information and System ScienceSeries, 1996. Haykin uses the common definition[A]⁺=(A^(H) A)⁻¹A^(H).  Equation 21where [A] is a rectangular matrix.

1. A method of communication using Orthogonal Frequency DivisionMultiplexing (‘OFDM”), the method comprising the steps of: generatingbit streams b_(n)∈(0,1),n=0,1, . . . ,K−1 and the corresponding sets offrequency domain carrier amplitudes (X₀(k) to X_(N)(k)), where k is theOFDM symbol number, modulated as OFDM symbols to be transmitted from atransmitter, inserting prefixes as guard intervals in said samplestreams, transmitting said OFDM symbols from said transmitter to areceiver, using information from said prefixes to estimate the ChannelImpulse Response (H_(D) ^((F))) of the transmission channels at thereceiver, and using the estimated Channel Impulse Response (Ĥ_(D)^((F))) to demodulate said bit streams in the signals received at saidreceiver, wherein said prefixes (α_(k)c₀ to α_(k)c_(D−1)) aredeterministic and are known to said receiver as well as to saidtransmitter.
 2. A method of communication as claimed in claim 1, whereinsaid prefixes (α_(k)c₀ to (α_(k)c_(D−1)) comprise a vector (P_(D)) thatis common to said symbols multiplied by at least one weighting factor(α_(k)).
 3. A method of communication as claimed in claim 2, whereinsaid weighting factor (α_(k)) differs from one symbol to another but theelements of a given vector (P_(D)) are multiplied by the same weightingfactor.
 4. A method of communication as claimed in claim 3, wherein saidweighting factor (α_(k)) has a pseudo-random value.
 5. A method ofcommunication as claimed in claim 1, wherein said weighting factor(α_(k)) is a complex value.
 6. A method of communication as claimed inclaim 5, wherein the modulus of said weighting factor (α_(k)) isconstant from one symbol to another.
 7. A method of communication asclaimed in claim 6, wherein said weighting factor (α_(k)) isproportional to ${\mathbb{e}}^{j\frac{2\pi}{N + D}m},$ where N is theuseful OFDM symbol size, D is the size of the prefix vector and m is aninteger.
 8. A method of communication as claimed claim 1, whereinestimating said Channel Impulse Response (H(F)/D) comprises performing aFourier Transform on a first vector (V_(HP)) that comprises the receivedsignal components corresponding to one of said prefixes (α_(k+1)c₀ toα_(k1)c_(D−1)) and also the received signal components corresponding tothe following one of said prefixes (α_(k)c₀ to α_(k)c_(D−1), α_(k+1)c₀to α_(k+1)c_(D−1)) to produce a known prefix transform (V_(P,F)), andperforming a component-by-component division of the receiving prefixsignal transform (V_(HP,F)) by known prefix transform (V_(P,F)).
 9. Amethod of communication s claimed in claim 8, wherein said prefixescomprise a vector (P_(D)) that is common to said symbols multiplied byweighting factors (α_(k,) α_(k+1)), said weighting factors differingfrom one symbol to another but the elements of a given vector beingmultiplied by the same weighting factor, and wherein the received signalcomponents corresponding to said one of said prefixes (α_(k)c₀ toα_(k)c_(D−1)) and said following one of said prefixes (α_(k+1)c₀ toα_(k+1)c_(D−)) are weighted by the respective value of said weightingfactor (α_(k,) α_(k+1)) before summing and performing said FourierTransform to produce said received prefix signal transform (V_(HP,F)).10. A method of communication as claimed in claim 8, wherein saidFourier Transforms are of dimension D×D, where D is the size of saidprefixes (c₀α_(k) to C_(D−1)α_(k)).
 11. A method of communication asclaimed in claim 8, wherein said Fourier Transforms are of dimension(D+N)×(D+N), where D is the size of said prefixes (α_(k)c₀ toα_(k)c_(D−1)) and N is thesize of the OFDM signals between saidprefixes, said first vector (V_(HP)) comprises said sum of said receivedsignal components corresponding to one of said prefixes (α_(k)c₀ toα_(k)c_(D−1)) and of the following one of said prefixes (α_(k+1)c₀ toα_(k+1)c_(D−1)) augmented by a zero value vector (0_(N) ^(T)) of size(N) to produce said received prefix signal transform (V_(HP,F)) of size(N+D), and said second vector (V_(P)) comprises said known components ofsaid prefixes ((α_(k)c₀ to α_(k)c_(D−1,) α_(k+1)c₀ to α_(k+1)c_(D−1))augmented by said zero value vector (0_(N) ^(T)) of size (N) to producesaid known prefix transform (V_(P,F)) of size (N+D).
 12. A method ofcommunication as claimed in any preceding claim 1, wherein estimatingsaid Channel Impulse Response (H_(D) ^((F))) comprises combininginformation from said prefixes (α_(k)c₀ to α_(k)c_(D−1,) α_(k+1)c₀ toα_(k+1)c_(D−1)) for more than one symbol to obtain said estimatedChannel Impulse Response (Ĥ_(D) ^((F))).
 13. A method of communicationas claimed in claim 1 wherein demodulating said bit streams comprises:performing the multiplication by a matrix proportional to${{R^{(1)}(k)} = {{\sqrt{N + D} \cdot \left\lbrack \hat{V} \right\rbrack \cdot r}\quad(k)}},{where}$$\begin{matrix}{\left\lbrack \hat{V} \right\rbrack = {\left( {{\sum\limits_{n = 0}^{N + D - 1}\quad{\beta_{k}}} - \frac{2n}{N + D}} \right)^{- \frac{1}{2}} \cdot}} \\{{{diag}\quad\left\{ {1,{\beta_{k}\frac{1}{N + D}},\ldots\quad,{\beta_{k}\frac{{N\_ D} - 1}{N + D}}} \right\}},}\end{matrix}$ ${\beta_{k} = {\frac{\alpha_{k}}{\alpha_{k + 1}}.}},$calculating the frequency shifted CIR coefficients${{\hat{H}}_{N + D}^{{Shifted},F} = \begin{pmatrix}{{\hat{H}\quad\left( {\beta_{k} - \frac{1}{N + D}} \right)\ldots}\quad,} \\{\hat{H}\quad\left( {\beta_{k} - {{\frac{1}{N + D} \cdot {\mathbb{e}}^{j\quad 2}}\frac{N + D - 1}{N + D}}} \right)}\end{pmatrix}},{\beta_{k} = {\frac{\alpha_{k}}{\alpha_{k + 1}}.}},$performing a component-by-component divisionR⁽²⁾(k) = R⁽¹⁾(k) = ⊗Ĥ_(N + D)^(Shifted, F), performing a multiplicationby a matrix proportional to${{R^{(3)}(k)} = {\left\lbrack \hat{V} \right\rbrack^{- 1} \cdot \frac{1}{\sqrt{N + D}} \cdot {{R^{(2)}(k)}.}}},$extracting the N equalized samples corresponding to the k^(th) datasymbol to the vector S^(EQ)(k), and transforming the symbol ŝ(k) intofrequency domain by performing a Fourier Transform:S _(F) ^(EQ)(k)=[F _(N×N) ]·S ^(EQ)(k).
 14. A method of communication sclaimed in claim 1, wherein demodulating said bit streams includespadding the received signal matrix and the operator matrices with zerosto obtain compatible dimensions for subsequent operations, multiplyingthe known prefix value matrix by the Channel Impulse Response estimationmatrix and subtracting the result from the received signal matrix.
 15. Atransmitter for use in a method of communication as claimed in claim 1and comprising generating means for generating bit streamsb_(n)∈(0,1),n=0,1 . . . ,K−1 modulated as OFDM symbols to be transmittedand inserting prefixes as guard intervals between said OFDM symbols,said prefixes (α_(k)c₀ to α_(k)c_(D−1)) being deterministic and suitableto be known to said receiver as well as to said transmitter.
 16. Areceiver for use in a method of communication as claimed in claim 1 andcomprising demodulating means for receiving signals that comprise bitstreams b_(n)∈(0,1),n=0,1 . . . ,K−1 modulated as OFDM symbols to betransmitted from a transmitter, with prefixes inserted in guardintervals from said transmitter to said receiver, said demodulatingmeans being arranged to use information from said prefixes to estimatethe Channel Impulse Response (H_(D) ^((F))) of the transmission channelsand to use the estimated Channel Impulse Response (Ĥ_(D) ^((F))) todemodulate said bit streams in the signals received at said receiver,said prefixes (α_(k)c₀ to α_(k)c_(D−1)) being deterministic and beingknown to said receiver as well as to said transmitter.